Is there an easy way to add a hole into an object?
Examples


In edit mode you can switch to face select mode (control tab).  Select faces and delete.  You may need to  created additional faces to delete.
Extrude or inset and remove faces
Use any mesh primitive with multiple faces and delete a face.  
Add more faces (loop cut and slide) and delete a face to create hole.
Use a Boolean modifier with operation Delete to create additional geometry suitable to the final step to delete faces.
In the image below we see from left to right


a plane
a plane with sphere placed to intersect the plane
goal mesh achieved with hole cut from Boolean modifier with [apply] clicked and user editing to delete extra vertices
plane selected with Boolean modifier present
plane with extra geometry created by Boolean modifier with [apply] clicked. Extra geometry will be deleted. N-gons may be present.






If rectangular outside with circular hole is desired use a torus or sphere or cylinder and use a shrink wrap to a rectangular outside.  Below is an example of a torus with 3 vertex cross section.





The geometry created above may be more suited to some situations rather than a triangulated n-gon.